full_name
stringlengths 3
121
| state
stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
| target_state
stringlengths 7
19k
| url
stringclasses 1
value | commit
stringclasses 1
value | file_path
stringlengths 21
79
|
|---|---|---|---|---|---|---|
Complex.tendsto_euler_sin_prod
|
z : ℂ
A :
Tendsto
(fun n =>
((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))
atTop (𝓝 (Complex.sin (↑π * z)))
⊢ 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
|
rw [<a>mul_one</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
Complex.tendsto_euler_sin_prod
|
case h.e'_3
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
Tendsto
(fun n =>
(↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))
atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
⊢ (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) =
fun n => (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)
|
ext1 n
|
case h.e'_3.h
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
Tendsto
(fun n =>
(↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))
atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
n : ℕ
⊢ (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) =
(∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
Complex.tendsto_euler_sin_prod
|
case h.e'_3.h
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
Tendsto
(fun n =>
(↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))
atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
n : ℕ
⊢ (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n) =
(∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)
|
congr 2 with x : 1
|
case h.e'_3.h.e_a.e_f.h
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
Tendsto
(fun n =>
(↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))
atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
n : ℕ
x : ℝ
⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
Complex.tendsto_euler_sin_prod
|
case h.e'_3.h.e_a.e_f.h
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
Tendsto
(fun n =>
(↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))
atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
n : ℕ
x : ℝ
⊢ Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n = ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)
|
rw [<a>mul_comm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
Complex.tendsto_euler_sin_prod
|
case h.e'_5
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A :
Tendsto
(fun n =>
(↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) *
((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) /
↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))))
atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))
⊢ 𝓝 1 = 𝓝 (Complex.cos (2 * z * ↑0))
|
rw [<a>Complex.ofReal_zero</a>, <a>MulZeroClass.mul_zero</a>, <a>Complex.cos_zero</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
|
isNilpotent_pi_of_bounded_class
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ Group.IsNilpotent ((i : η) → Gs i)
|
rw [<a>nilpotent_iff_lowerCentralSeries</a>]
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ ∃ n, lowerCentralSeries ((i : η) → Gs i) n = ⊥
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Nilpotent.lean
|
isNilpotent_pi_of_bounded_class
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ ∃ n, lowerCentralSeries ((i : η) → Gs i) n = ⊥
|
refine ⟨n, ?_⟩
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ lowerCentralSeries ((i : η) → Gs i) n = ⊥
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Nilpotent.lean
|
isNilpotent_pi_of_bounded_class
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ lowerCentralSeries ((i : η) → Gs i) n = ⊥
|
rw [<a>eq_bot_iff</a>]
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ lowerCentralSeries ((i : η) → Gs i) n ≤ ⊥
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Nilpotent.lean
|
isNilpotent_pi_of_bounded_class
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ lowerCentralSeries ((i : η) → Gs i) n ≤ ⊥
|
apply <a>le_trans</a> (<a>lowerCentralSeries_pi_le</a> _)
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ (pi Set.univ fun i => lowerCentralSeries (Gs i) n) ≤ ⊥
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Nilpotent.lean
|
isNilpotent_pi_of_bounded_class
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ (pi Set.univ fun i => lowerCentralSeries (Gs i) n) ≤ ⊥
|
rw [← <a>eq_bot_iff</a>, <a>Subgroup.pi_eq_bot_iff</a>]
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ ∀ (i : η), lowerCentralSeries (Gs i) n = ⊥
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Nilpotent.lean
|
isNilpotent_pi_of_bounded_class
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
⊢ ∀ (i : η), lowerCentralSeries (Gs i) n = ⊥
|
intro i
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
i : η
⊢ lowerCentralSeries (Gs i) n = ⊥
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Nilpotent.lean
|
isNilpotent_pi_of_bounded_class
|
G : Type u_1
inst✝³ : Group G
H : Subgroup G
inst✝² : H.Normal
η : Type u_2
Gs : η → Type u_3
inst✝¹ : (i : η) → Group (Gs i)
inst✝ : ∀ (i : η), Group.IsNilpotent (Gs i)
n : ℕ
h : ∀ (i : η), Group.nilpotencyClass (Gs i) ≤ n
i : η
⊢ lowerCentralSeries (Gs i) n = ⊥
|
apply lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (h i)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Nilpotent.lean
|
List.modifyLast_append_one
|
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a : α
l : List α
⊢ modifyLast f (l ++ [a]) = l ++ [f a]
|
cases l with | <a>List.nil</a> => simp only [<a>List.nil_append</a>, <a>List.modifyLast</a>, <a>List.modifyLast.go</a>, <a>Array.toListAppend_eq</a>, <a>Array.data_toArray</a>] | <a>List.cons</a> _ tl => simp only [<a>List.cons_append</a>, <a>List.modifyLast</a>] rw [<a>List.modifyLast.go</a>] case x_3 => exact <a>List.append_ne_nil_of_ne_nil_right</a> tl [a] (<a>List.cons_ne_nil</a> a []) rw [<a>List.modifyLast.go_append_one</a>, <a>Array.toListAppend_eq</a>, <a>Array.push_data</a>, <a>Array.data_toArray</a>, <a>List.nil_append</a>, <a>List.cons_append</a>, <a>List.nil_append</a>, <a>List.cons_inj_right</a>] exact modifyLast_append_one _ _ tl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
List.modifyLast_append_one
|
case nil
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a : α
⊢ modifyLast f ([] ++ [a]) = [] ++ [f a]
|
simp only [<a>List.nil_append</a>, <a>List.modifyLast</a>, <a>List.modifyLast.go</a>, <a>Array.toListAppend_eq</a>, <a>Array.data_toArray</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
List.modifyLast_append_one
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast f (head✝ :: tl ++ [a]) = head✝ :: tl ++ [f a]
|
simp only [<a>List.cons_append</a>, <a>List.modifyLast</a>]
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (head✝ :: (tl ++ [a])) #[] = head✝ :: (tl ++ [f a])
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
List.modifyLast_append_one
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (head✝ :: (tl ++ [a])) #[] = head✝ :: (tl ++ [f a])
|
rw [<a>List.modifyLast.go</a>]
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (tl ++ [a]) (#[].push head✝) = head✝ :: (tl ++ [f a])
case cons.x_3
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ tl ++ [a] = [] → False
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
List.modifyLast_append_one
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (tl ++ [a]) (#[].push head✝) = head✝ :: (tl ++ [f a])
case cons.x_3
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ tl ++ [a] = [] → False
|
case x_3 => exact <a>List.append_ne_nil_of_ne_nil_right</a> tl [a] (<a>List.cons_ne_nil</a> a [])
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (tl ++ [a]) (#[].push head✝) = head✝ :: (tl ++ [f a])
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
List.modifyLast_append_one
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (tl ++ [a]) (#[].push head✝) = head✝ :: (tl ++ [f a])
|
rw [<a>List.modifyLast.go_append_one</a>, <a>Array.toListAppend_eq</a>, <a>Array.push_data</a>, <a>Array.data_toArray</a>, <a>List.nil_append</a>, <a>List.cons_append</a>, <a>List.nil_append</a>, <a>List.cons_inj_right</a>]
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (tl ++ [a]) #[] = tl ++ [f a]
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
List.modifyLast_append_one
|
case cons
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ modifyLast.go f (tl ++ [a]) #[] = tl ++ [f a]
|
exact modifyLast_append_one _ _ tl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
List.modifyLast_append_one
|
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
f : α → α
a head✝ : α
tl : List α
⊢ tl ++ [a] = [] → False
|
exact <a>List.append_ne_nil_of_ne_nil_right</a> tl [a] (<a>List.cons_ne_nil</a> a [])
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Basic.lean
|
AddLECancellable.lt_add_of_tsub_lt_left
|
α : Type u_1
β : Type u_2
inst✝³ : PartialOrder α
inst✝² : AddCommSemigroup α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hb : AddLECancellable b
h : a - b < c
⊢ a < b + c
|
rw [<a>lt_iff_le_and_ne</a>, ← <a>tsub_le_iff_left</a>]
|
α : Type u_1
β : Type u_2
inst✝³ : PartialOrder α
inst✝² : AddCommSemigroup α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hb : AddLECancellable b
h : a - b < c
⊢ a - b ≤ c ∧ a ≠ b + c
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/Sub/Defs.lean
|
AddLECancellable.lt_add_of_tsub_lt_left
|
α : Type u_1
β : Type u_2
inst✝³ : PartialOrder α
inst✝² : AddCommSemigroup α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hb : AddLECancellable b
h : a - b < c
⊢ a - b ≤ c ∧ a ≠ b + c
|
refine ⟨h.le, ?_⟩
|
α : Type u_1
β : Type u_2
inst✝³ : PartialOrder α
inst✝² : AddCommSemigroup α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hb : AddLECancellable b
h : a - b < c
⊢ a ≠ b + c
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/Sub/Defs.lean
|
AddLECancellable.lt_add_of_tsub_lt_left
|
α : Type u_1
β : Type u_2
inst✝³ : PartialOrder α
inst✝² : AddCommSemigroup α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hb : AddLECancellable b
h : a - b < c
⊢ a ≠ b + c
|
rintro rfl
|
α : Type u_1
β : Type u_2
inst✝³ : PartialOrder α
inst✝² : AddCommSemigroup α
inst✝¹ : Sub α
inst✝ : OrderedSub α
b c d : α
hb : AddLECancellable b
h : b + c - b < c
⊢ False
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/Sub/Defs.lean
|
AddLECancellable.lt_add_of_tsub_lt_left
|
α : Type u_1
β : Type u_2
inst✝³ : PartialOrder α
inst✝² : AddCommSemigroup α
inst✝¹ : Sub α
inst✝ : OrderedSub α
b c d : α
hb : AddLECancellable b
h : b + c - b < c
⊢ False
|
simp [hb] at h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/Sub/Defs.lean
|
mul_inv_le_of_le_mul
|
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
m n : ℕ
hab : a ≤ b * c
⊢ a * c⁻¹ ≤ b
|
by_cases h : c = 0
|
case pos
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
m n : ℕ
hab : a ≤ b * c
h : c = 0
⊢ a * c⁻¹ ≤ b
case neg
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
m n : ℕ
hab : a ≤ b * c
h : ¬c = 0
⊢ a * c⁻¹ ≤ b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/GroupWithZero/Canonical.lean
|
mul_inv_le_of_le_mul
|
case pos
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
m n : ℕ
hab : a ≤ b * c
h : c = 0
⊢ a * c⁻¹ ≤ b
|
simp [h]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/GroupWithZero/Canonical.lean
|
mul_inv_le_of_le_mul
|
case neg
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
m n : ℕ
hab : a ≤ b * c
h : ¬c = 0
⊢ a * c⁻¹ ≤ b
|
exact <a>le_of_le_mul_right</a> h (by simpa [h] using hab)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/GroupWithZero/Canonical.lean
|
mul_inv_le_of_le_mul
|
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
m n : ℕ
hab : a ≤ b * c
h : ¬c = 0
⊢ a * c⁻¹ * c ≤ b * c
|
simpa [h] using hab
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/GroupWithZero/Canonical.lean
|
NNReal.one_le_coe
|
r r₁ r₂ : ℝ≥0
x y : ℝ
⊢ 1 ≤ ↑r ↔ 1 ≤ r
|
rw [← <a>NNReal.coe_le_coe</a>, <a>NNReal.coe_one</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/NNReal/Basic.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
⊢ L.distinctConstantsTheory s =
⋃ t, L.distinctConstantsTheory ↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) t)
|
simp only [<a>FirstOrder.Language.distinctConstantsTheory</a>]
|
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
⊢ (fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) '' (s ×ˢ s ∩ (diagonal α)ᶜ) =
⋃ t,
(fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) ''
(↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) t) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) t) ∩
(diagonal α)ᶜ)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
⊢ (fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) '' (s ×ˢ s ∩ (diagonal α)ᶜ) =
⋃ t,
(fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) ''
(↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) t) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) t) ∩
(diagonal α)ᶜ)
|
rw [← <a>Set.image_iUnion</a>, ← <a>Set.iUnion_inter</a>]
|
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
⊢ (fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) '' (s ×ˢ s ∩ (diagonal α)ᶜ) =
(fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) ''
((⋃ i,
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i)) ∩
(diagonal α)ᶜ)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
⊢ (fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) '' (s ×ˢ s ∩ (diagonal α)ᶜ) =
(fun ab => ((L.con ab.1).term.equal (L.con ab.2).term).not) ''
((⋃ i,
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i)) ∩
(diagonal α)ᶜ)
|
refine congr(_ '' ($(?_) ∩ _))
|
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
⊢ s ×ˢ s =
⋃ i,
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
⊢ s ×ˢ s =
⋃ i,
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i)
|
ext ⟨i, j⟩
|
case h.mk
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
⊢ (i, j) ∈ s ×ˢ s ↔
(i, j) ∈
⋃ i,
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
case h.mk
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
⊢ (i, j) ∈ s ×ˢ s ↔
(i, j) ∈
⋃ i,
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i) ×ˢ
↑(Finset.map (Function.Embedding.subtype fun x => x ∈ s) i)
|
simp only [<a>Set.prod_mk_mem_set_prod_eq</a>, <a>Finset.coe_map</a>, <a>Function.Embedding.coe_subtype</a>, <a>Set.mem_iUnion</a>, <a>Set.mem_image</a>, <a>Finset.mem_coe</a>, <a>Subtype.exists</a>, <a>Subtype.coe_mk</a>, <a>exists_and_right</a>, <a>exists_eq_right</a>]
|
case h.mk
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
⊢ i ∈ s ∧ j ∈ s ↔ ∃ i_1, (∃ (x : i ∈ s), ⟨i, ⋯⟩ ∈ i_1) ∧ ∃ (x : j ∈ s), ⟨j, ⋯⟩ ∈ i_1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
case h.mk
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
⊢ i ∈ s ∧ j ∈ s ↔ ∃ i_1, (∃ (x : i ∈ s), ⟨i, ⋯⟩ ∈ i_1) ∧ ∃ (x : j ∈ s), ⟨j, ⋯⟩ ∈ i_1
|
refine ⟨fun h => ⟨{⟨i, h.1⟩, ⟨j, h.2⟩}, ⟨h.1, ?_⟩, ⟨h.2, ?_⟩⟩, ?_⟩
|
case h.mk.refine_1
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
h : i ∈ s ∧ j ∈ s
⊢ ⟨i, ⋯⟩ ∈ {⟨i, ⋯⟩, ⟨j, ⋯⟩}
case h.mk.refine_2
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
h : i ∈ s ∧ j ∈ s
⊢ ⟨j, ⋯⟩ ∈ {⟨i, ⋯⟩, ⟨j, ⋯⟩}
case h.mk.refine_3
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
⊢ (∃ i_1, (∃ (x : i ∈ s), ⟨i, ⋯⟩ ∈ i_1) ∧ ∃ (x : j ∈ s), ⟨j, ⋯⟩ ∈ i_1) → i ∈ s ∧ j ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
case h.mk.refine_1
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
h : i ∈ s ∧ j ∈ s
⊢ ⟨i, ⋯⟩ ∈ {⟨i, ⋯⟩, ⟨j, ⋯⟩}
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
case h.mk.refine_2
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
h : i ∈ s ∧ j ∈ s
⊢ ⟨j, ⋯⟩ ∈ {⟨i, ⋯⟩, ⟨j, ⋯⟩}
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
case h.mk.refine_3
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
⊢ (∃ i_1, (∃ (x : i ∈ s), ⟨i, ⋯⟩ ∈ i_1) ∧ ∃ (x : j ∈ s), ⟨j, ⋯⟩ ∈ i_1) → i ∈ s ∧ j ∈ s
|
rintro ⟨t, ⟨is, _⟩, ⟨js, _⟩⟩
|
case h.mk.refine_3.intro.intro.intro.intro
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
t : Finset ↑s
is : i ∈ s
h✝¹ : ⟨i, ⋯⟩ ∈ t
js : j ∈ s
h✝ : ⟨j, ⋯⟩ ∈ t
⊢ i ∈ s ∧ j ∈ s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
case h.mk.refine_3.intro.intro.intro.intro
L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
n : ℕ
s : Set α
i j : α
t : Finset ↑s
is : i ∈ s
h✝¹ : ⟨i, ⋯⟩ ∈ t
js : j ∈ s
h✝ : ⟨j, ⋯⟩ ∈ t
⊢ i ∈ s ∧ j ∈ s
|
exact ⟨is, js⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/ModelTheory/Syntax.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
|
have h_sum_a : ∀ a, <a>Summable</a> fun n => f n a := fun a => Pi.summable.mpr fun b => <a>ENNReal.summable</a>
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
|
have h_sum : <a>Summable</a> fun n => f n := Pi.summable.mpr h_sum_a
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
⊢ withDensity κ (∑' (n : ι), f n) = kernel.sum fun n => withDensity κ (f n)
|
ext a s hs
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ((withDensity κ (∑' (n : ι), f n)) a) s = ((kernel.sum fun n => withDensity κ (f n)) a) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ((withDensity κ (∑' (n : ι), f n)) a) s = ((kernel.sum fun n => withDensity κ (f n)) a) s
|
rw [<a>ProbabilityTheory.kernel.sum_apply'</a> _ a hs, <a>ProbabilityTheory.kernel.withDensity_apply'</a> κ _ a s]
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
|
swap
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
|
have : ∫⁻ b in s, (∑' n, f n) a b ∂κ a = ∫⁻ b in s, ∑' n, (fun b => f n a b) b ∂κ a := by congr with b rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a a)]
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
|
rw [this, <a>MeasureTheory.lintegral_tsum</a> fun n => (<a>Measurable.of_uncurry_left</a> (hf n)).<a>Measurable.aemeasurable</a>]
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∑' (i : ι), ∫⁻ (a_1 : β) in s, f i a a_1 ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∑' (i : ι), ∫⁻ (a_1 : β) in s, f i a a_1 ∂κ a = ∑' (n : ι), ((withDensity κ (f n)) a) s
|
congr with n
|
case h.h.e_f.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
n : ι
⊢ ∫⁻ (a_1 : β) in s, f n a a_1 ∂κ a = ((withDensity κ (f n)) a) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h.h.e_f.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
n : ι
⊢ ∫⁻ (a_1 : β) in s, f n a a_1 ∂κ a = ((withDensity κ (f n)) a) s
|
rw [<a>ProbabilityTheory.kernel.withDensity_apply'</a> _ (hf n) a s]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
|
have : <a>Function.uncurry</a> (∑' n, f n) = ∑' n, <a>Function.uncurry</a> (f n) := by ext1 p simp only [<a>Function.uncurry_def</a>] rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a _), <a>tsum_apply</a>] exact Pi.summable.mpr fun p => <a>ENNReal.summable</a>
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (Function.uncurry (∑' (n : ι), f n))
|
rw [this]
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (∑' (n : ι), Function.uncurry (f n))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
⊢ Measurable (∑' (n : ι), Function.uncurry (f n))
|
exact <a>Measurable.ennreal_tsum'</a> hf
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ Function.uncurry (∑' (n : ι), f n) = ∑' (n : ι), Function.uncurry (f n)
|
ext1 p
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Function.uncurry (∑' (n : ι), f n) p = tsum (fun n => Function.uncurry (f n)) p
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Function.uncurry (∑' (n : ι), f n) p = tsum (fun n => Function.uncurry (f n)) p
|
simp only [<a>Function.uncurry_def</a>]
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ tsum (fun n => f n) p.1 p.2 = tsum (fun n p => f n p.1 p.2) p
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ tsum (fun n => f n) p.1 p.2 = tsum (fun n p => f n p.1 p.2) p
|
rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a _), <a>tsum_apply</a>]
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Summable fun n p => f n p.1 p.2
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
p : α × β
⊢ Summable fun n p => f n p.1 p.2
|
exact Pi.summable.mpr fun p => <a>ENNReal.summable</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
|
congr with b
|
case e_f.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
b : β
⊢ tsum (fun n => f n) a b = ∑' (n : ι), (fun b => f n a b) b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
ProbabilityTheory.kernel.withDensity_tsum
|
case e_f.h
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : ↥(kernel α β)
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : ↥(kernel α β)
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
b : β
⊢ tsum (fun n => f n) a b = ∑' (n : ι), (fun b => f n a b) b
|
rw [<a>tsum_apply</a> h_sum, <a>tsum_apply</a> (h_sum_a a)]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/WithDensity.lean
|
MultilinearMap.exists_bound_of_continuous
|
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
cases <a>isEmpty_or_nonempty</a> ι
|
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : IsEmpty ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
case inr
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
obtain ⟨ε : ℝ, ε0 : 0 < ε, hε : ∀ m : ∀ i, E i, ‖m - 0‖ < ε → ‖f m - f 0‖ < 1⟩ := <a>NormedAddCommGroup.tendsto_nhds_nhds</a>.1 (hf.tendsto 0) 1 <a>zero_lt_one</a>
|
case inr.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m - 0‖ < ε → ‖f m - f 0‖ < 1
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m - 0‖ < ε → ‖f m - f 0‖ < 1
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
simp only [<a>sub_zero</a>, f.map_zero] at hε
|
case inr.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
rcases <a>NormedField.exists_one_lt_norm</a> 𝕜 with ⟨c, hc⟩
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
have : 0 < (‖c‖ / ε) ^ <a>Fintype.card</a> ι := <a>pow_pos</a> (<a>div_pos</a> (zero_lt_one.trans hc) ε0) _
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
refine ⟨_, this, ?_⟩
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
⊢ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
⊢ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖
|
refine f.bound_of_shell_of_continuous hf (fun _ => ε0) (fun _ => hc) fun m hcm hm => ?_
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ ‖f m‖ ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ ‖f m‖ ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖
|
refine (hε m ((<a>pi_norm_lt_iff</a> ε0).2 hm)).le.trans ?_
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ 1 ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ 1 ≤ (‖c‖ / ε) ^ Fintype.card ι * ∏ i : ι, ‖m i‖
|
rw [← <a>div_le_iff'</a> this, <a>one_div</a>, ← <a>inv_pow</a>, <a>inv_div</a>, <a>Fintype.card</a>, ← <a>Finset.prod_const</a>]
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ ∏ _x : ι, ε / ‖c‖ ≤ ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inr.intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ ∏ _x : ι, ε / ‖c‖ ≤ ∏ i : ι, ‖m i‖
|
exact <a>Finset.prod_le_prod</a> (fun _ _ => <a>div_nonneg</a> ε0.le (<a>norm_nonneg</a> _)) fun i _ => hcm i
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : IsEmpty ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
|
refine ⟨‖f 0‖ + 1, <a>add_pos_of_nonneg_of_pos</a> (<a>norm_nonneg</a> _) <a>zero_lt_one</a>, fun m => ?_⟩
|
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : IsEmpty ι
m : (i : ι) → E i
⊢ ‖f m‖ ≤ (‖f 0‖ + 1) * ∏ i : ι, ‖m i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : IsEmpty ι
m : (i : ι) → E i
⊢ ‖f m‖ ≤ (‖f 0‖ + 1) * ∏ i : ι, ‖m i‖
|
obtain rfl : m = 0 := <a>funext</a> (<a>IsEmpty.elim</a> ‹_›)
|
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : IsEmpty ι
⊢ ‖f 0‖ ≤ (‖f 0‖ + 1) * ∏ i : ι, ‖0 i‖
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
MultilinearMap.exists_bound_of_continuous
|
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)
inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SeminormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : MultilinearMap 𝕜 E G
hf : Continuous ⇑f
h✝ : IsEmpty ι
⊢ ‖f 0‖ ≤ (‖f 0‖ + 1) * ∏ i : ι, ‖0 i‖
|
simp [<a>Finset.univ_eq_empty</a>, <a>zero_le_one</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
TruncatedWittVector.commutes'
|
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
m : ℕ
hm : n ≤ m
x : ZMod (p ^ m)
⊢ ((truncate hm).comp (zmodEquivTrunc p m).toRingHom) x = (zmodEquivTrunc p n) ((ZMod.castHom ⋯ (ZMod (p ^ n))) x)
|
rw [<a>TruncatedWittVector.commutes</a> _ _ hm]
|
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
m : ℕ
hm : n ≤ m
x : ZMod (p ^ m)
⊢ ((zmodEquivTrunc p n).toRingHom.comp (ZMod.castHom ⋯ (ZMod (p ^ n)))) x =
(zmodEquivTrunc p n) ((ZMod.castHom ⋯ (ZMod (p ^ n))) x)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/WittVector/Compare.lean
|
TruncatedWittVector.commutes'
|
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
m : ℕ
hm : n ≤ m
x : ZMod (p ^ m)
⊢ ((zmodEquivTrunc p n).toRingHom.comp (ZMod.castHom ⋯ (ZMod (p ^ n)))) x =
(zmodEquivTrunc p n) ((ZMod.castHom ⋯ (ZMod (p ^ n))) x)
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/WittVector/Compare.lean
|
Filter.pi_neBot
|
ι : Type u_1
α : ι → Type u_2
f f₁ f₂ : (i : ι) → Filter (α i)
s : (i : ι) → Set (α i)
p : (i : ι) → α i → Prop
⊢ (pi f).NeBot ↔ ∀ (i : ι), (f i).NeBot
|
simp [<a>Filter.neBot_iff</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Pi.lean
|
SimpleGraph.incidenceFinset_eq_filter
|
V : Type u_1
G : SimpleGraph V
e : Sym2 V
v : V
inst✝² : Fintype ↑(G.neighborSet v)
inst✝¹ : DecidableEq V
inst✝ : Fintype ↑G.edgeSet
⊢ G.incidenceFinset v = filter (Membership.mem v) G.edgeFinset
|
ext e
|
case a
V : Type u_1
G : SimpleGraph V
e✝ : Sym2 V
v : V
inst✝² : Fintype ↑(G.neighborSet v)
inst✝¹ : DecidableEq V
inst✝ : Fintype ↑G.edgeSet
e : Sym2 V
⊢ e ∈ G.incidenceFinset v ↔ e ∈ filter (Membership.mem v) G.edgeFinset
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Combinatorics/SimpleGraph/Finite.lean
|
SimpleGraph.incidenceFinset_eq_filter
|
case a
V : Type u_1
G : SimpleGraph V
e✝ : Sym2 V
v : V
inst✝² : Fintype ↑(G.neighborSet v)
inst✝¹ : DecidableEq V
inst✝ : Fintype ↑G.edgeSet
e : Sym2 V
⊢ e ∈ G.incidenceFinset v ↔ e ∈ filter (Membership.mem v) G.edgeFinset
|
induction e
|
case a.h
V : Type u_1
G : SimpleGraph V
e : Sym2 V
v : V
inst✝² : Fintype ↑(G.neighborSet v)
inst✝¹ : DecidableEq V
inst✝ : Fintype ↑G.edgeSet
x✝ y✝ : V
⊢ s(x✝, y✝) ∈ G.incidenceFinset v ↔ s(x✝, y✝) ∈ filter (Membership.mem v) G.edgeFinset
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Combinatorics/SimpleGraph/Finite.lean
|
SimpleGraph.incidenceFinset_eq_filter
|
case a.h
V : Type u_1
G : SimpleGraph V
e : Sym2 V
v : V
inst✝² : Fintype ↑(G.neighborSet v)
inst✝¹ : DecidableEq V
inst✝ : Fintype ↑G.edgeSet
x✝ y✝ : V
⊢ s(x✝, y✝) ∈ G.incidenceFinset v ↔ s(x✝, y✝) ∈ filter (Membership.mem v) G.edgeFinset
|
simp [<a>SimpleGraph.mk'_mem_incidenceSet_iff</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Combinatorics/SimpleGraph/Finite.lean
|
Monoid.fg_iff_submonoid_fg
|
M : Type u_1
N✝ : Type u_2
inst✝¹ : Monoid M
inst✝ : AddMonoid N✝
N : Submonoid M
⊢ FG ↥N ↔ N.FG
|
conv_rhs => rw [← N.range_subtype, <a>MonoidHom.mrange_eq_map</a>]
|
M : Type u_1
N✝ : Type u_2
inst✝¹ : Monoid M
inst✝ : AddMonoid N✝
N : Submonoid M
⊢ FG ↥N ↔ (Submonoid.map N.subtype ⊤).FG
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Finiteness.lean
|
Monoid.fg_iff_submonoid_fg
|
M : Type u_1
N✝ : Type u_2
inst✝¹ : Monoid M
inst✝ : AddMonoid N✝
N : Submonoid M
⊢ FG ↥N ↔ (Submonoid.map N.subtype ⊤).FG
|
exact ⟨fun h => h.out.map N.subtype, fun h => ⟨h.map_injective N.subtype <a>Subtype.coe_injective</a>⟩⟩
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Finiteness.lean
|
Finset.mk_mem_sym2_iff
|
α : Type u_1
s t : Finset α
a b : α
⊢ s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s
|
rw [<a>Finset.mem_mk</a>, <a>Finset.sym2_val</a>, <a>Multiset.mk_mem_sym2_iff</a>, <a>Finset.mem_mk</a>, <a>Finset.mem_mk</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Finset/Sym.lean
|
MvPolynomial.monic_monomial_eq
|
R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
n m✝ : σ
s : σ →₀ ℕ
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S₁
p q : MvPolynomial σ R
m : σ →₀ ℕ
⊢ (monomial m) 1 = m.prod fun n e => X n ^ e
|
simp [<a>MvPolynomial.monomial_eq</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/MvPolynomial/Basic.lean
|
exists_bounded_mem_Icc_of_closed_of_le
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
hx : x ∈ s
⊢ (BoundedContinuousFunction.const X a + (b - a) • f) x = Function.const X a x
|
simp [hfs hx]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UrysohnsBounded.lean
|
exists_bounded_mem_Icc_of_closed_of_le
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
hx : x ∈ t
⊢ (BoundedContinuousFunction.const X a + (b - a) • f) x = Function.const X b x
|
simp [hft hx]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UrysohnsBounded.lean
|
exists_bounded_mem_Icc_of_closed_of_le
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
⊢ a ≤ (BoundedContinuousFunction.const X a + (b - a) • f) x
|
dsimp
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
⊢ a ≤ a + (b - a) * f x
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UrysohnsBounded.lean
|
exists_bounded_mem_Icc_of_closed_of_le
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
⊢ a ≤ a + (b - a) * f x
|
nlinarith [(hf01 x).1]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UrysohnsBounded.lean
|
exists_bounded_mem_Icc_of_closed_of_le
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
⊢ (BoundedContinuousFunction.const X a + (b - a) • f) x ≤ b
|
dsimp
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
⊢ a + (b - a) * f x ≤ b
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UrysohnsBounded.lean
|
exists_bounded_mem_Icc_of_closed_of_le
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : NormalSpace X
s t : Set X
hs : IsClosed s
ht : IsClosed t
hd : Disjoint s t
a b : ℝ
hle : a ≤ b
f : X →ᵇ ℝ
hfs : EqOn (⇑f) 0 s
hft : EqOn (⇑f) 1 t
hf01 : ∀ (x : X), f x ∈ Icc 0 1
x : X
⊢ a + (b - a) * f x ≤ b
|
nlinarith [(hf01 x).2]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/UrysohnsBounded.lean
|
MeasureTheory.Measure.map_toOuterMeasure
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f μ
⊢ (map f μ).toOuterMeasure = ((OuterMeasure.map f) μ.toOuterMeasure).trim
|
rw [← <a>MeasureTheory.Measure.trimmed</a>, <a>MeasureTheory.OuterMeasure.trim_eq_trim_iff</a>]
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f μ
⊢ ∀ (s : Set β), MeasurableSet s → (map f μ).toOuterMeasure s = ((OuterMeasure.map f) μ.toOuterMeasure) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_toOuterMeasure
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f μ
⊢ ∀ (s : Set β), MeasurableSet s → (map f μ).toOuterMeasure s = ((OuterMeasure.map f) μ.toOuterMeasure) s
|
intro s hs
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
hf : AEMeasurable f μ
s : Set β
hs : MeasurableSet s
⊢ (map f μ).toOuterMeasure s = ((OuterMeasure.map f) μ.toOuterMeasure) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_toOuterMeasure
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
hf : AEMeasurable f μ
s : Set β
hs : MeasurableSet s
⊢ (map f μ).toOuterMeasure s = ((OuterMeasure.map f) μ.toOuterMeasure) s
|
simp [hf, hs]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
ContinuousMultilinearMap.completeSpace
|
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
|
have H : ∀ {m : Π i, E i}, <a>Continuous</a> fun f : (Π i, E i) →ᵤ[{s | <a>Bornology.IsVonNBounded</a> 𝕜 s}] F ↦ <a>UniformOnFun.toFun</a> _ f m := (<a>UniformOnFun.uniformContinuous_eval</a> (<a>Bornology.isVonNBounded_covers</a>) _).<a>UniformContinuous.continuous</a>
|
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
ContinuousMultilinearMap.completeSpace
|
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
|
rw [<a>completeSpace_iff_isComplete_range</a> uniformEmbedding_toUniformOnFun.toUniformInducing, <a>ContinuousMultilinearMap.range_toUniformOnFun</a>]
|
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsComplete
{f |
Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
(∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
(toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
(toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
(toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
ContinuousMultilinearMap.completeSpace
|
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsComplete
{f |
Continuous ((toFun {s | IsVonNBounded 𝕜 s}) f) ∧
(∀ (m : (i : ι) → E i) (i : ι) (x y : E i),
(toFun {s | IsVonNBounded 𝕜 s}) f (update m i (x + y)) =
(toFun {s | IsVonNBounded 𝕜 s}) f (update m i x) + (toFun {s | IsVonNBounded 𝕜 s}) f (update m i y)) ∧
∀ (m : (i : ι) → E i) (i : ι) (c : 𝕜) (x : E i),
(toFun {s | IsVonNBounded 𝕜 s}) f (update m i (c • x)) =
c • (toFun {s | IsVonNBounded 𝕜 s}) f (update m i x)}
|
simp only [<a>Set.setOf_and</a>, <a>Set.setOf_forall</a>]
|
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsComplete
({a | Continuous ((toFun {s | IsVonNBounded 𝕜 s}) a)} ∩
((⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 + i_3)) =
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_2) +
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)}) ∩
⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 • i_3)) =
i_2 • (toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)}))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
ContinuousMultilinearMap.completeSpace
|
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsComplete
({a | Continuous ((toFun {s | IsVonNBounded 𝕜 s}) a)} ∩
((⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 + i_3)) =
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_2) +
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)}) ∩
⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 • i_3)) =
i_2 • (toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)}))
|
apply_rules [<a>IsClosed.isComplete</a>, <a>IsClosed.inter</a>]
|
case h₁
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsClosed {a | Continuous ((toFun {s | IsVonNBounded 𝕜 s}) a)}
case h₂.h₁
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsClosed
(⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 + i_3)) =
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_2) +
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)})
case h₂.h₂
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsClosed
(⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 • i_3)) =
i_2 • (toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)})
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
ContinuousMultilinearMap.completeSpace
|
case h₁
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsClosed {a | Continuous ((toFun {s | IsVonNBounded 𝕜 s}) a)}
|
exact <a>UniformOnFun.isClosed_setOf_continuous</a> h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
ContinuousMultilinearMap.completeSpace
|
case h₂.h₁
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsClosed
(⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 + i_3)) =
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_2) +
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)})
|
exact <a>isClosed_iInter</a> fun m ↦ <a>isClosed_iInter</a> fun i ↦ <a>isClosed_iInter</a> fun x ↦ <a>isClosed_iInter</a> fun y ↦ <a>isClosed_eq</a> H (H.add H)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
ContinuousMultilinearMap.completeSpace
|
case h₂.h₂
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : UniformSpace F
inst✝⁴ : UniformAddGroup F
inst✝³ : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : CompleteSpace F
inst✝ : T2Space F
h : RestrictGenTopology {s | IsVonNBounded 𝕜 s}
H : ∀ {m : (i : ι) → E i}, Continuous fun f => (toFun {s | IsVonNBounded 𝕜 s}) f m
⊢ IsClosed
(⋂ i,
⋂ i_1,
⋂ i_2,
⋂ i_3,
{x |
(toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 (i_2 • i_3)) =
i_2 • (toFun {s | IsVonNBounded 𝕜 s}) x (update i i_1 i_3)})
|
exact <a>isClosed_iInter</a> fun m ↦ <a>isClosed_iInter</a> fun i ↦ <a>isClosed_iInter</a> fun c ↦ <a>isClosed_iInter</a> fun x ↦ <a>isClosed_eq</a> H (H.const_smul _)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
|
riemannZeta_eulerProduct
|
s : ℂ
hs : 1 < s.re
⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (riemannZeta s))
|
rw [← <a>tsum_riemannZetaSummand</a> hs]
|
s : ℂ
hs : 1 < s.re
⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean
|
riemannZeta_eulerProduct
|
s : ℂ
hs : 1 < s.re
⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n))
|
apply <a>EulerProduct.eulerProduct_completely_multiplicative</a> <| <a>summable_riemannZetaSummand</a> hs
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean
|
dist_pointReflection_right
|
α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : NormedAddCommGroup W
inst✝⁴ : MetricSpace Q
inst✝³ : NormedAddTorsor W Q
𝕜 : Type u_6
inst✝² : NormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
p q : P
⊢ dist ((Equiv.pointReflection p) q) q = ‖2‖ * dist p q
|
simp [<a>dist_eq_norm_vsub</a> V, <a>Equiv.pointReflection_vsub_right</a> (G := V), <a>nsmul_eq_smul_cast</a> 𝕜, <a>norm_smul</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
Real.sign_apply_eq
|
r : ℝ
⊢ r.sign = -1 ∨ r.sign = 0 ∨ r.sign = 1
|
obtain hn | rfl | hp := <a>lt_trichotomy</a> r (0 : ℝ)
|
case inl
r : ℝ
hn : r < 0
⊢ r.sign = -1 ∨ r.sign = 0 ∨ r.sign = 1
case inr.inl
⊢ sign 0 = -1 ∨ sign 0 = 0 ∨ sign 0 = 1
case inr.inr
r : ℝ
hp : 0 < r
⊢ r.sign = -1 ∨ r.sign = 0 ∨ r.sign = 1
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Real/Sign.lean
|
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